ex Tables for Statisticians and Biometricians [XVIII XXII 



This gives a range of + '587 to - '993. Chauvenet's test would throw out the 

 five negative and the three positive extreme individuals as anomalous, although 

 + '587 is near to -f- '62, and again really gives the + '62 as doubtful. 



(7) Proceeding lastly to Irwin's test we take first the gap between '73 and 

 - 1-38, and find \ = '65/-24357 = 2'67 nearly. If such a gap had been between the 

 second and third the chance of its occurrence would be < '001, and our chance 

 between fifth and sixth must be far less than this. This leads to rejection. Now 

 take the positive end. Here we have 



X,= -45/-24357 = l-85, 



and we find that the chance must be less than '001, of such a gap occurring. Hence 

 according to Irwin's test the doubtful observation '62 as well as the two beyond it 

 should be rejected. These results only mean that the causes which are producing 

 the colour-vision of the centre or normal part of the frequency differ widely from 

 those at the tails. The individuals at the tails have anomalous colour-vision. 



It must be remembered of course that all three criteria are based on the 

 assumption that the parent population i$ approximately of normal type. 



TABLES XXI, XXI bis AND XXII. 



The Distribution of the Extreme Individuals and of the Range in Samples from 

 a Normal Population. (L. H. C. Tippett, Biometrika, Vol. xvn. pp. 364387; E. S. 

 Pearson, Biometrika, Vol. xvm. pp. 173 194; "Student," Biometrika, Vol. xix. 

 pp. 151164.) 



1. Suppose that a sample of n is drawn from a univariate normal distribution 

 and that the character is measured from the population mean in terms of the popu- 

 lation standard deviation as unit. Let u and v be the largest and smallest values of 

 the character found in the sample, so that the sample range is given by w = u v. 

 Then if the distribution of w in repeated samples be y =f(u), the probability integral 

 of this curve is 



f /()&*-{*(!+)}" (1), 



J -oo 



where ^(1 + M ) is found by entering Sheppard's tables of the normal curve with 

 x = u. The distribution for v is the same but reversed. 



Table XXI, which was calculated by Tippett, gives the expression (1) for various 

 values of n and u. Diagram 1 shows those values of the extreme variate, u (or v), the 

 chances of exceeding which in random sampling are (a) '05 and (b) '01. 



2. Table XXI bis (first published here) is an extension by E. S. Pearson of one 

 given by Tippett*. It shows the deviation from the population mean, measured 

 in terms of the population standard deviation as unit, which will only be exceeded 

 by the extreme variate in (1) 10%, (2) 5%, (3) 1 % and (4) O5 % of random 



* Biometrika, Vol. xvn. p. 267. 



